Generalizing the notion of Koszul algebra

Abstract

We introduce a generalization of the notion of a Koszul algebra, which includes graded algebras with relations in different degrees, and we establish some of the basic properties of these algebras. This class is closed under twists, twisted tensor products, regular central extensions and Ore extensions. We explore the monomial algebras in this class and we include some well-known examples of algebras that fall into this class.

Generalization of quasi-Koszul algebras

“…This property can be fully reflected by the δ-Koszul algebra [4]. Motivated by Cassidy and Shelton's work [2], we tried another way to generalize piecewise-Koszul algebra by omitting the purity requirement and got the concept of K p algebra in our paper [12]. All of the algebras mentioned above are graded algebras.…”

The Yoneda Algebra of a 𝒦2Algebra Need not be Another 𝒦2Algebra

“…We use the algorithm given in section 5 of [2] to prove that A is K 2 . From the set R one can calculate that S 1 = {m, n, p, q, r, s, t, u, v, w, x, y, z}, S 2 = {mn 2 , n 2 pq, npqr, pqrs, st, tuvw, uvwx, vwxy, xy 2 }, S 3 = {pqr, vw}, S 4 = {n 2 } and S 5 = ∅.…”

“…The following natural generalization of Koszul was introduced in [2] and also investigated in [7] and [8]. We write E n (A) for p Ext n,m A (K, K).…”

Skew categories, smash product categories and quasi-Koszul categories

“…There exist numerous generalizing notations of Koszul algebras. For example, Berger [2] introduced the notation of N -Koszul algebras, and Cassidy and Shelton [3] introduced the notation of K 2 algebras. Very recently, Martínez-Villa and Solberg [7] introduced the notation of (weakly, quasi-)Koszul categories to obtain a naturally associated Koszul theory for any finite dimensional algebra.…”